Discrete Mathematics Course Overview

Discrete Mathematics

• Discrete mathematics is the study of mathematical structures that are countable or otherwise distinct and separable.
• It deals with objects that can consider only distinct, separated values.
• Examples: number of students, houses, integer numbers, etc.

Importance of Discrete Mathematics

• Computer is built on discrete structures.
• Foundation of logic building and problem solving.
• Understanding algorithms and their complexities.
• Software Development.

Propositional Logic

• A proposition is a statement that is either true or false, but not both.
• A proposition is a declarative sentence that has the following properties:
  • It can be either true or false.
  • It cannot be neither.
  • It cannot be both.
• Examples:
  • "The sky is blue" is a proposition because it can be clearly classified as true or false.
  • "4 + 4 = 5" is a proposition because it can be identified as false.

Propositions and Variables

• A proposition can be represented by a variable or symbol.
• Example: P = "Earth is a Planet" and Q = "Earth is a food".
• A proposition can be combined or modified with logical connectives/linkers.

Logical Connectives

• Linkers:
  • And (∧)
  • Or (∨)
  • If (→)
  • If and only if (⇔)
  • Negation / NOT (¬)
• Negation (¬):
  • True if the actual proposition is false.
  • Example: ¬p = "Sakib is not a batsman".

Conjunction and Disjunction

• Conjunction (AND):
  • True if both propositions are true.
  • Example: p ∧ q = "Tamim Iqbal is a batsman and Mustafizur Rahman is a bowler".
• Disjunction (OR):
  • True if any proposition is true.
  • Example: p ∨ q = "Tamim Iqbal is a batsman or Mustafizur Rahman is a bowler".

Exclusive Disjunction (XOR)

• True if exactly one proposition is true.
• Example: p ⊕ q = "Quinton de Kock will play in today's match or Heinrich Klassen will play in today's match".

Truth Tables

• A truth table is used to observe the behavior of a compound proposition at a glance.
• It shows the different possible truth values of the simple propositions and the truth value of the compound one.
• Example: (p ∨ q) ∧ ¬r.

Solving Compound Propositions

• Divide the compound proposition into smaller sections.
• Use the truth table to find the truth value of each section.
• Combine the results to find the final truth value of the compound proposition.